Hi PHil
The most common 2 are Shopify and WordPress. Both are great for Search Engine Optimisation been around a long time, integrate with a lot of other software and reasonably easy to find the developers for them. However they have different cost/fee structures, pros and cons.

Generally, Shopify is a proprietry software, so if you want a high level of customisation, it is difficult as well as if you keep adding extensions, you keep paying every month. So it can add up if you have a number of extensions. However the user interface is very friendly, hosting etc all taken care of and you can do most things yourself.

While WordPress user interface does not appear as friendly as Shopify however once you get used to it, it is easy.
It is also an open source, so if you have a number of plugins, often you pay a one-off fees and that’s it, no ongoing fees.
You can also customise the site in every possible way as there are hundreds of themes and it can scale to a reasonably good extend.
On the flipside- it really depends on the WordPress developer you select.
A good developer knows how to choose the right theme for you, build a robust site, sort out the hosting and most importantly ensure the backend interface they select is easy for you to use (without locking key elements). On the otherhand, if you get a bad one- it can be clunky and frustrating.

There are also other e-commerce platforms eg RocketSpark but I do not know enough about them to comment.
Please stay away from Wix and Weebly- they are cheap/ free but not right for e-commerce.

Hope that this helps.
thanks
Anne